,
1398
EVERSOLE, PETERSON AND KINDSVATER
DIFFUSION COEFFICIENTS I N SOLUTION: AK' IRIPROVED METHOD FOR CALCULATING D AS A FUNCTIOK OF CONCENTRATION' W. G. EVERSOLE, J. DONALD PETERSON, AND H. M. KINDSVATER Division of Physical Chemistry, State University of Iowa, Iowa City,Iowa Received June 9 , 1041
I n previous papers (1, 2) a new method was described for the determination of diffusion coefficients in solution, and the method was applied to the measurement of the diffusion of methylene blue in gelatin gels, The method consists of the measurement by means of a slit photometer of the concentration a t various levels in a diffusion column, as diffusion progresses from a fixed concentration a t the zero level. The experimental method is accurate and relatively easy and rapid, and permits the determination of the diffusion coefficient over a range of concentrations from a single experiment. However, the method of calculation used is open to objection: and it was the purpose of this work to derive an exact equation for calculating diffusion coefficients from the results of measurements by this method. THEORY
The equation derived by Eversole and Doughty (I) for this calculation was
m = m e -zBl4Dt
(1)
mo is the (constant) concentration a t the zero level, and m is the concentration a t the x level a t time t since the beginning of the diffusion process. The mathematical approximation was introduced in the derivation in passing from equation 8 in the previous paper (l),which refers to a diminishing concentration a t the zero level, to equation 1 above. For purposes of calculation, equation 1 was transformed to give
1 This work was aided by a grant from the Graduate College of the State University of Iowa. * Private communication from Dr. J. Norton Wilson, California Institute of Technology, Pasadena, California. The authors wish t o thank Dr. Wilson for his criticism and for the stimulating discussion by correspondence, which led indirectly to the development of the improved method of calculation presented here.
1399
DIFFUSIOK COEFFICIENTS IN SOLCTIOK
de
and D was obtained from the slope of the curve obtained by plotting against x ( t constant). This gives a curve for each value of t .
I t would have been preferable t o plot
TABLE 1 Digusion of methylene blue i n 6 per cent gelatin gels (mo = 1000 mg. per liter) I
m
I
B
0 10 30 50
500 600
lwm
dE X 10'
1
+
p*
I
,
D X
106
cm.2 per accord
0 3.93 12.49 20.44 32.43 40.79
m
80
400
10'
cm. aec.-liZ
mg. per Mer
100 200 300
x
2.885 2.387 2.183 2.000 1.907 1.606 1.377 1.173 1.002 0.870
1,000 1.1361 1.1744 1.1873 1.2027 1.2139 1.26% 1,3432 1.4540 1.5778 1.6934
84.44 141.79 213.03 289.46 361.98
1.635 1.858 1.920 1.941 1,966 1.985 2.065 2.196 2.377 2.580 2.769
so that a single curve would have been obtained. Actually, this curve is very nearly linear. Eversole and Doughty (2) assumed that it was exactly linear in accordance with equation 2 and therefore d0
(3)
m The magnitude of the error involved in calculating D by means of equation 3 may be seen from table l. However, in common with many other diffusion equations, equation 1 implicitly assumes that D is independent of concentration. Some workers have placed themselves in the position of showing how D varies with concentration by means of calculations made by the use of equations which are mathematically invalid unless D is independent of concentration. I n the following derivation it is assumed that the diffusion coefficient ( D J is a function of concentration.
1400
EVERBOLE, PETERSON AND KINDSVATER
Fisk's law for linear diffusion, where D,varies with concentration, may
be writtens
For convenience, C = m/nio, where m and mo have t.he same significance
as in the previous eqmtions. Let
and
(3 =
and
Substituting equations 6 and 8 in equation 4)
-D
- [;(
- - - D-C
3
1
$
(9)
Since C is expressed entirely in terms of D, and 9, we may remove the restrictions on the partial derivatives and write
pcgd(Dc!?$) =
-lc
29dC
(10)
since, when C = 0, dC/d9 = 0. Integrating and solving for D c ,
a = -2dBdC p
0
C
Equation 11 may be integrated by parts to give
D,= 8
-E dC (9C -
This may be written in the usual form
C:)*-.@,
only if D,is assumed to be constant.
1' CdB)
(11)
DIFFUSION COEFFICIENTS I N SOLUTION
since, when C = 0, e =
p
1401
; or
Equation 13 differs from equation 3 by the factor in the brackets. This factor approaches 1 as C approaches 0,and therefore it may be concluded that equation 3 is a batisfactory approximation if C < < 1 even if the curve is not strictly linear, as awimed by Eversole and Doughty (2). For making calculations, equation 13 map be transformed to give
I n order to evaluate the second favtor in eqnation 14, it is necessary to ___ plot the experimental values of 0 against - In C. The determination of the slope a t the various concentrations is greatly facilitated by the fact that the rurve is very nmrly linear. The integral in the third factor m w t also be evaluated graphically from the curve obtained by plotting C against 8. The effect of errors inherent in graphical integration with a limit of % arc largely minimized by the fact thaL C approaches zero rapidly with increase in 0 and, a t the lowest concentrations whwe the percentage uncertainty in the integral is greatest, the term involving the integral is least important in determining the value of D,. ChLCULATIONS
Since it W M found by Eversale __ __and Doughty (2) that the curve obtained by plotting e against t/ - In C was a straight line passing through thc origin, their results for the diffusion of methylene blue can be recalculated most simply by combining equations 3 and 14, using their value of D (1.635 X l e 6cm.2 per second), and substituting C = m/m+
D, = 1.635 X
(1
+ em
ni de) B
Figure 1 is a plot of m (= lOOOC) against 0, using their data. of
e, nil and
imm
The values
dB in table 1 were obtained a t round concentrations from
an enlarged drawing of figure 1. The values of D,were obtained by substitution in equation 15. It will be observed that D,will always increase with concentration in any concentration range where the relation between O2 and In C is linear. However, if the slope of this curvc is greater at the lower concentrations, D, may remain constant or decrease with concentration. A plot of D,against the square root of the concentration (milligrams per liter) is shown in figure 2. The rapid rise in D,in the dilute range is some-
1402
EVERSOLE, PETERSON AND KINDSVATER
what surprising. This may result from the assumption of a linear relationship between O2 and In C. However, this curve is so nearly linear that it is difficult to see how a large error could be made in determining the slope by graphical means. There is an error of constant magnitude in all
0
200
m
4 00
10
mg p r r b t e r
6 00
600
FIG. 1. Diffusion data for methylene blue
..F I.0 0
50
I50
10.0
ZOO
250
mf
FIG.2. Diffiision coefficients of methylene blue
of the integrals as a result of the uncertainty involved in integrating to infinity. However, an error of 10 per cent of the integral for the lowest concentration would give an error in Do of only 1 per cent a t this concentration, decreasing to about 0.04 per cent at the highest concentration.
EFFECT OF DIELECTRIC COSSTAST ON ISVERSION
OF SUCROSE
1403
While this method is not ideal, since it involves graphical computations, it permits the calculation of D, over a range of concentrations from the results of a single experiment. I t should be particularly useful where the diffusion coefficient changes rapidly with concentration, as in dilute salt solutions. SUMMARY
1. The method proposed by Eversole and Doughty for the calculation of diffusion coefficients is in error except a t the lowest conecntrations. 2. A new equation is derived which expresses D as a function of concentration for all concent>rations. REFERESCES (1) EVERSOLE, W. G., ASD DOUGHTY, E. W.: J. Phys. Chem. 39, 289 (1935). (2) EVERSOLE, W.G., . ~ X DDOUOHTY, E. IT.: ,J, Phys. Chem. 41, 663 (1937).
T H E RATE OF IXVERSIOK OF SUCROSE AS A F U S C T I O S O F T H E DIELECTRIC COSSTAKT OF T H E SOI,IyE?;T' CHARLES J. PLASK
AND
HERSCHEL IIUS'r
D e p n r h c n f of Chemislry, P u r d u e r n i c e r s i l y , W e s t LQfoyeltc, I n d i a n a Received J u n e 13, 1941 IKTRODUCTIOS
The reaction used in this study is distinguished by being the first one to be timed (24). Since Wilhelniy did his work in 1850, an unparalleled number of measurements have been made in connection with this change. It is not, however, the inversion of sucrose in itself which is of interest here, but rather, a much more general problem, the effect' of the solvent upon reaction rates in solution. Our understanding of the kinetics of gaseous reactions is in a far more satisfactory state than our knowledge of reactions in solution. This is exactly what would be expected, owing both to t,he fact that the reaction is taking place in the liquid state and even more to the presence of an extremely large number of solvent molecules. The reaction rate constant may be reprcscntcd by an equation of thc ,same general type as thc cnipirical Arrhenius equation (2), both for gaseous rcactions and for reactions in solution. However, in the latter This article is based upon a thesis submitted by Charles J . Plank t o the Faculty of P,irdue Cniversity in partial fulfillmcnt of the requircmcnts for th(h degrcc of Doctor of Philosophy, Junc, 1942.
